1 – Introduction In this article, we show that the problem (Pλ,f) (∆(∆u) =λ|u|qc−2u+f in Ω, u= ∆u= 0 on∂Ω, (1.1) where Ω is a smooth bounded domain inRN, N >4

ON NONHOMOGENEOUS BIHARMONIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT

M. Guedda

Abstract: In this paper we consider the problem ∆²u = λ|u|^qc−2u+f in Ω, u = ∆u = 0 on ∂Ω, where qc = 2N/(N−4), N > 4, is the limiting Sobolev expo- nent and Ω is a smooth bounded domain in R^N. Under some restrictions on f and λ, the existence of weak solutionuis proved. Moreoveru≥0 forf ≥0 wheneverλ≥0.

1 – Introduction

In this article, we show that the problem (P_λ,f)

(∆(∆u) =λ|u|^qc−2u+f in Ω,

u= ∆u= 0 on∂Ω,

(1.1)

where Ω is a smooth bounded domain inR^N, N >4, ∆ is the Laplacian operator and q_c = 2N/(N−4), has weak solutions inH_θ²(Ω) =H²(Ω)∩H₀¹(Ω) equipped with the norm

kuk_H²

θ = µZ

Ω|∆u|²

¶1/2

. To this end we consider the functional

F_λ(u) = 1 2 Z

Ω

|∆u|²dx− λ qc

Z

Ω

|u|^qcdx− Z

Ω

f u dx , u∈H_θ²(Ω), λ >0. (1.2)

Under some suitable conditions, it is proved that (1.1) admits at least two solu- tions. Our arguments make use of the mountain pass theorem and of the Lions concentration-compactness principle.

Received: November 10, 1997; Revised: January 15, 1998.

AMS(MOS) Subject Classification: 35J65, 35J20, 49J45.

Recently, Van der Vorst [10] considered the following problem S = inf

½Z

Ω

|∆u|²; u∈H_θ²(Ω), Z

Ω

|u|^qc = 1

¾ . (1.3)

He proved that the infimum in (1.3) is never achieved by a function u∈ H_θ²(Ω) when Ω is bounded. In contrast Hadiji, Picard and the author in [7] considered the problem

Sϕ = inf

½Z

Ω|∆u|²; u∈H_θ²(Ω), Z

Ω|u+ϕ|^qc = 1

¾ . (1.4)

They showed that the infimum in (1.4) is achieved wheneverϕis continuous and non identically equal to zero. More precisely it is shown that, for any minimizing sequence (um) for (1.4), there exists a subsequence (umk) and a function u ∈ H_θ²(Ω) such that

u_mk * u weakly inH_θ²(Ω) and ku+ϕk_qc = 1 .

On the other hand, Bernis et al. [1] considered a variant of (1.1) where f is replaced by β|u|^p−2u, 1< p <2. They proved the existence of at least two positive solutions forβ sufficiently small. At this stage, we would like to mention that when Ω =R^N P.L. Lions [9] proved that S is achieved only by the function uε defined by

u_ε(x) =

h(N −4) (N−2)N(N+ 2)ε²ⁱ

N−4 8

³ε+|x−a|^2´

N−4 2

, x∈R^N ,

for anya∈R^N and anyε >0. This note is organized as follows. In Section 2 we verify thatF_λ satisfies the (PS)_c condition. In Section 3 we prove the existence of a local minimizeru ofF_λ. Moreover, we show that u≥0 wheneverf ≥0 and λ≥0. Section 4 is devoted to the existence of a second solution to (1.4). The results presented in this paper have been announced in [6].

Notice that if f ≡ 0, the result of Section 3 is valid and gives the trivial solutionu= 0. The method we adopt is closely related to the one of [3].

Before the verification of the (PS)_c condition, let us remark that if v is a solution to (1.1) thenu=λ^qc1−2 v satisfies

(∆(∆u) =|u|^qc−2u+g in Ω,

u= ∆u= 0 on∂Ω,

(1.5)

whereg=λ^qc1−2f.

2 – Verification of the (PS)_c condition

Let Ω be a bounded domain in R^N, N > 4, and f ∈ L²(Ω). We denote by F_λ: H_θ²(Ω)→Rthe functional defined by

F_λ(u) = 1 2 Z

Ω

|∆u|²dx− λ q_c

Z

Ω

|u|^qcdx− Z

Ω

f u dx , (2.1)

where ∆ is the Laplacian operator and λis a real parameter. We first look for critical points ofF ^def=F₁. We show that F satisfies the Palais–Smale condition in a suitable sublevel strip.

Let S be the best Sobolev embedding constant of H_θ²(Ω) intoL^qc(Ω); that is S = inf

½Z

Ω

|∆u|²; u∈H_θ²(Ω), Z

Ω

|u|^qc = 1

¾ (2.2)

and

K= N^qcq

2q (4q_c)^qcq kfk^q_q, q = q_c q_c−1 . (2.3)

Proposition 2.1. The functional F satisfies the (P S)_c condition in the sublevel strip(−∞, _N² S^N4−K); that is if {u_m}is a sequence inH_θ²(Ω)such that

F(um)→c and dF(um)→0 inH_θ⁻²(Ω), (2.4)

where

c < 2

N S^N4 −K ,

then{u_m} contains a subsequence which converges strongly inH_θ²(Ω).

Proof: Let{u_m}be a sequence inH_θ²(Ω) which satisfies (2.4). From (2.4) it is easy to see that{u_m}is bounded inH_θ²(Ω); thus there is a subsequence{u_mk}, and an elementu of H_θ²(Ω) such that

u_mk * u weakly in H_θ²(Ω) (2.5)

and

umk →u strongly in L^p(Ω), 1≤p < qc and a.e. in Ω. (2.6)

The concentration-compactness Lemma of Lions [9] asserts the existence of at most a countable index setJ and positive constants {ν_j},j∈J such that

|umk|^qc *|u|^qc+^X

j∈J

νjδxj , (2.7)

weakly in the sense of measures, and

|∆u_mk|²→µ , (2.8)

for some positive bounded measureµ. Moreover, µ ≥ |∆u|²+^X

j∈J

S ν

N−4

jN δ_xj , (2.9)

where

x_j ∈Ω and ν_j = 0 or ν_j ≥S^N4 . (2.10)

We assert thatν_j = 0 for eachj. If not, assume thatν_j0 6= 0,for some j₀. From the hypothesis (2.4),

c= lim

k→∞F(u_mk)−1 2

DdF(u_mk), u_mk^E , c≥ 2

N Z

Ω

|u|^qc−1 2

Z

Ω

f u+ 2 N S^N4 . Using the H¨older inequality one has

c ≥ 2

N S^N4 − N^qcq

2q (4q_c)^qcq kfk^q_q .

This contradicts the hypothesis. Consequentlyν_j = 0 for each j and

k→∞lim Z

Ω|umk|^qc = Z

Ω|u|^qc , which implies

u_mk →u strongly in H_θ²(Ω). The proof is complete.

3 – Existence of a solution

In this part we consider the problem of finding solutions to (P_λ,f).We show, under suitable conditions onf andλ, thatF_λ has an infimum on a small ball in H_θ²(Ω). We suppose first that λ = 1, and denote by F the functional F₁. The proof is based on the following lemma.

Lemma 3.1. There exist constants r and R > 0 such that if kfk₂ ≤ R, then

F(u)≥0 for all kuk_H²

θ(Ω) =r . (3.1)

Proof: Thanks to the Sobolev and H¨older inequalities we have F(u)≥ 1

2 Z

Ω

|∆u|²− 1 qc

S^−qc³ Z

Ω

|∆u|^2´

qc

2−|Ω|^12−qc1 S⁻¹kfk₂^³ Z

Ω

|∆u|^2´1/2. (3.2)

Inequality (3.2) can be written

F(u)≥h^³kuk_H²

θ

´, (3.3)

where h(x) = 1

2x²−λ0x^qc −λ1x , λ0 = 1

q_c S^−qc and λ1=kfk2|Ω|

1 2−¹

qc S⁻¹ . Let

g(x) = 1

2x−λ₀x^qc−1−λ₁ for x≥0 .

There existsλ >0 such that, if 0< λ1 ≤λ, g attains its positive maximum and we get (3.1), with

r=

µq_cS^qc 2

¶ ¹

qc−1

and R=|Ω|^−12+qc1 S λ , thanks to (3.3).

Remark 3.1. Arguing as above we can see that there exists a constantα >0 such that

F(u)≥α, for all kuk_H²

θ =r .

Proposition 3.1. LetR andr be given by Lemma 3.1. Suppose thatf 6≡0 and

max^³kfk₂,kfk_q^´<min(R⁰, R) , (3.4)

where

R⁰= 4q_cS^4qN N^³2 (q_c−1)^´

1 q

. Then there exists a functionu₁ ∈H_θ²(Ω)such that

F(u₁) = min

Br

F(v)<0 , (3.5)

where

Br=ⁿv ∈H_θ², kvk_H²

θ(Ω) < r^o,

andu₁ is a solution to (P_1,f).Moreover,u₁ ≥0 whenever f ≥0.

Proof: Without loss of generality, we can suppose that f(a) >0 for some a∈Ω.

Let

u_²(x) = ε^N−44 φ(x)

³ε+|x−a|^2´

N−4 2

, ε >0 ,

where φ ∈ C₀^∞(Ω) is a fixed function such that 0 ≤ φ ≤ 1 and φ ≡ 1 in some neighbourhood ofa.

Since

Z

Ωf u²dx >0, for a small ε , we can chooset >0 sufficiently small such that

F(t u_²)<0 . Hence

infBr

F(v)<0 . (3.6)

Let{u_m}be a minimizing sequence of (3.6). From (3.4) and Lemma 3.1 we may assume that

ku_mk_H²

θ < r₀< r . (3.7)

According to the Ekeland variational principle [5] we may assume

∆²u_m− |u_m|^qc−f → 0 in H_θ⁻²(Ω). (3.8)

On the other hand, from (2.3) and (3.4), we get 1

N S^N4 −K >0 . (3.9)

We deduce, from (3.8)–(3.9) and Proposition 2.1, that {u_m} has a subsequence converging tou₁ ∈H_θ²(Ω), and u₁ is a weak solution to (P_1,f).

Now we suppose that f ≥ 0. Let v ∈ H_θ²(Ω) be a solution to the following problem

−∆v=|∆u₁|.

As in [10, 11] we getv >0,v≥ |u₁|in Ω, Z

Ω

|∆v|²= Z

Ω

|∆u₁|² and Z

Ω

|v|^qc ≥ Z

Ω

|u₁|^qc . It then follows that

F(v)≤F(u1) and kvk_H²

θ ≤r . ConsequentlyF is minimized by a positive function.

This method allows us under suitable conditions on f and λ, to prove the existence of solutions to (P_λ,f).

Theorem 3.1. Suppose that f 6≡ 0, then there exists λ_f > 0 such that if the following condition is satisfied

0< λ_f < λ^qc1−2 <min µ 1

kfk₂, 1 kfk_q

¶

min(R⁰, R), (3.10)

Problem(P)_λ,f has at least one solutionu_λ. Moreover u_λ ≥0 whenever f ≥0.

Proof: For the proof we consider Problem (P_1,g) where g = f λ^qc1−2. Condition (3.10) implies that g satisfies (3.4). So the existence follows imme- diately from Proposition 3.1.

Now suppose, on the contrary, that u_λ exists for anyλsuch that 0< λ^qc1−2 <min

µ 1 kfk₂, 1

kfk_q

¶

min(R⁰, R) .

Note that, sinceλ^−qc1−2 u_λ is the solution to (P_1,g) obtained by (3.5), we have ku_λk_H²

θ(Ω) ≤ r λ^qc1−2 . It follows from this thatku_λk_H²

θ(Ω) →0 asλ↓0.

Passing to the limit in (P_λ,f) we deduce that f ≡0, which yields to a contra- diction.

4 – Existence of a second solution

In this section we shall show, under additional conditions that (P_λ,f) possesses a second solution. Here we use the mountain pass theorem without the Palais–

Smale condition [2, 8]. As in the preceding section, we first deal with the caseλ= 1.

Assume that condition (3.4) is satisfied and that f >0 in some neighbourhood ofa. Set

v_ε= u_ε kuεkqc

. The main result of this section is the following.

Theorem 4.1. There exists t₀>0such that if f satisfies kfk^q_q < t₀

K1

Z

Ω

f v_εdx, for small enough ε >0 , (4.1)

where

K₁ = N^qcq 2q (4q_c)^qcq , then(P_1,f) has at least two distinct solutions.

Proof: The proof relies on a variant of the mountain pass theorem without the (PS) condition. We have, forεsufficiently small (see [4]),

k∆v_εk²₂ = S+ O(ε^N−42 ). (4.2)

Set

h(t) = F(t v_ε) = 1

2t²X_ε− 1

q_c t^qc−t Z

Ω

f v_εdx for t≥0 , whereX_ε =k∆v_εk²₂.

Sinceh(t) goes to−∞astgoes to +∞, sup_t≥0h(t) is achieved at somet_ε ≥0.

Remark 3.1 asserts thatt_ε>0, and we deduce h⁰(t_ε) =t_ε(X_ε−t^q_ε^c−2)−

Z

Ω

f v_εdx= 0 and h⁰⁰(t_ε)≤0 , (4.3)

thus

µ 1 q_c−1

¶ ¹

qc−2

X

1 qc−2

ε ≤ t_ε ≤ X

1 qc−2

ε .

(4.4)

Lett0= ¹₂(_q¹

c−1)^qc1−2S^qc1−2. We deduce from (4.2) and (4.4) that, for ε0 small, t₀ < t_ε for ε∈(0, ε₀) .

(4.5) Thus

sup

t≥0

h(t) = sup

t≥t0

h(t) .

On the other hand, since the functiont −→ ¹₂t²X_ε− _q¹

c t^qc is increasing on the interval [0, X

1 qc−2

ε ], we get h(tε) ≤ 2

N S^N4 −tε

Z

Ωf vεdx+ O(ε^N−42 ) , thanks to (4.2). Hence

h(t_ε) ≤ 2

N S^N4 −t₀ Z

Ω

f v_εdx+ O(ε^N−42 ) . (4.6)

Consequently if we let

t₀ Z

Ωf v_εdx > K₁kfk^q_q , (4.7)

we deduce that

sup

t≥0

F(tv_ε) < 2

N S^N4 −K . (4.8)

Note that there existst₁ large enough such that F(t1vε)<0 and kt1vεk_H²

θ > r , (4.9)

wherer is given by Lemma 3.1. Hence α≤c2 = inf

γ∈Γ max

s∈[0,1]F(γ(s))< 2

N S^N4 −K , where

Γ =

½

γ ∈C^³[0,1], H_θ²(Ω)^´: γ(0) = 0, γ(1) =t₁v_ε

¾ ,

provided ε is small enough. Then, according to the mountain pass theorem without the (PS) condition, there exists a sequence{u_m} inH_θ²(Ω) such that

F(u_m)→c₂ and dF(u_m)→0 inH_θ⁻²(Ω).

Sincec₂ < _N² S^N4 −K, we deduce from Proposition 2.1 that there exists u₂ such thatc₂ =F(u₂) andu₂ is a weak solution to (P_1,f).

This solution is distinct fromu₁ sincec₁<0< c₂. So the proof is complete.

Finally, by using Theorem 4.1, we deduce the Corollary 4.1. Assume (3.10). If

λ^qcq−1−2 < t0

K₁kfk^qq

Z

Ωf vεdx ,

forεsmall enough, then problem(P_λ,f) has at least two solutions.

ACKNOWLEDGEMENTS– The author thanks C. Picard and M. Kirane for their useful discussions and for informing him of Ref. [1] and the referee for careful examination of the paper and valuable remarks. This work was partially supported by C.C.I. K´enitra (Maroc).

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M. Guedda,

LAMFA, Facult´e de Math´ematiques et d’Informatique, Universit´e de Picardie Jules Verne, 33, rue Saint-Leu, 80039 Amiens – FRANCE